The simulation problem for high order linear differential systems

The central object of interest of this paper are systems of linear constant coefficient ordinary differential equations of arbitrary order of the form s(d/dt)w = M(d/dt)f with G and M given, but otherwise arbitrary, polynomial matrices. In these equations w and f are vector-valued functions of which f is assumed to be given, while w is the solution to (1) that we are looking for. Alongside (1) we also consider initial conditions obtained by imposing the value at time t = 0 of a linear combination of the variables w and their derivatives, yielding s(d/dt)_W(0)=Ta with S a polynomial matrix, T a fixed real matrix, and a a real vector. The three main issues we will address concerning such equations are: i) Solvability, meaning the problem of finding conditions that assure that a solution w to R (d/dt) w = M (d/dt) f exists for a (or for any) given vector distribution f. ii) Determining the index, which, as we shall see, means investigating how smoothness of the given function f is related to smoothness of the solution w. iii) Compatibility of initial conditions, in which case the initial conditions S (d/dt) w(0) = Ta are considered alongside equations R (d/dt) w = M (d/dt) f. The problem then becomes one of first checking when S (d/dt) w is well defined, and then providing conditions under which R(d/dt)w = M (d/dt) f admit a (unique) solution w that satisfies also S (d/dt) w(0) = Ta for a (or for any) given a. We show how our results generalize to equations of arbitrary order classical properties of first order systems such as E d/dt w + Aw = f.